Maclaurin polynomial error bounds

Question

I need some help with my Calculus II Maclaurin polynomial error bounds.

$Mn(x)$ is the $n^{th}$ Maclaurin polynomial for $f(x) = e^x$. I need to use the error bound formula to determine a value of $n$ such that $|Mn(2) − e^2| < 10^{−4}$.

Thanks in advance for a helpful explanation of how to solve this one!

Answer 1

$f(x) = e^x\\ f_n(x) = 1 + x + \frac {x^2}{2} + \cdots + \frac {x^n}{n!}\\ |f_n(x) - f(x)|= \epsilon_n\\ \epsilon_n < \frac {f^{(n+1)}(\xi)}{(n+1)!}(x)^{n+1}$

in the interval $[0,2]$

$\epsilon_n < \frac {e^2}{(n+1)!}2^{n+1} < 10^{-4}\\ \frac {(n+1)!}{2^{n+1}}> (10^4)(e^2) = 73,891$

if $n>9$ then $f_n(2) -e^2 < 10^{-4}$